Rekenrek (Eng Version

7/22/2019 Rekenrek (Eng Version

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REKENREK

A Resource for Teachers

A Tool for Children





Sekilas Pandang

• Dalam sesi ini kita akan menguji Rekenrek,benda yg ringkas, tapi berkuasa untuk membina

kemahiran berfikir dan merangsang pemahaman

Matematik.

• Khususnya , kita akan melihat… – Rasional penggunaan Rekenrek

– Rekenrek dalam Matematik

– Aktiviti Rekenrek yang boleh meningkatkan kemahiran

berfikir murid;kefahaman dan kepakaran menambahdan menolak nombor berdasarkan sistem asas

sepuluh.



Pengenalan

• Solve th.• As you come up with the answer, be aware of the

strategy that you used to determine the answer.

• Most likely, your brain will make adjustments on

these numbers very quickly, and you will use aninformal strategy to find the result. Of course,some of you will know these by rotememorization as well.



8 + 7 = ?

• What mental adjustments did you make as you

solved this problem?

– Double 8, subtract 1? (8 + 8 = 16; 16 - 1 = 15)

– Double 7, add 1? (7 + 7 = 14; 14 + 1 = 15)

– Make 10, add 5? (8 + 2 = 10; 10 + 5 = 15)

– Make 10 another way? (7 + 3 = 10)

– Other strategies?

• Next problem…



5 + 8 = ?

• What mental adjustments did you make as yousolved this problem? – Make 10, add 3? (5 + 5 = 10; 10 + 3 = 13)

– Make 10 another way? (8 + 2 = 10; 10 + 3 = 13)

– Use another fact? (If 8 + 4 = 12, then 8 + 5 = 13)


• Next problem…



9 + 7 = ?

• What mental adjustments did you make as yousolved this problem? – Make 10, add 6? (9 + 1 = 10; 10 + 6 = 16)




Questions… • If we use these strategies as adults, do we teach

them explicitly to young children?• Should we?

• If so… how?

With the Rekenrek • The Rekenrek is a powerful tool that helps children

subitize (see “inside” numbers), develop cardinality(one-to-one correspondence), work flexibly with

numbers by using decomposition (part-part-whole)strategies, anchors of 5 and 10, and informalstrategies s.a. doubling, halving, one/two more, andone/two less.



What is the Rekenrek?

• Developed by Adri Treffers• The rekenrek combines key features of othermanipulative models like counters, the numberline, and base-10 models.

• It is comprised of two strings of 10 beads each,strategically broken into groups of five.

• The rekenrek therefore entices students to thinkin groups of 5 and 10.

• The structure of the rekenrek offers visual picturesfor young learners, encouraging them to “see”numbers within other numbers… to see groups of5 and 10.

• For example…



Visual Model• With the rekenrek, young learners learn quickly

to “see” the number 7 in two distinct parts: One

group of 5, and 2 more.

A group of 10

3 more

• Similarly, 13 is seen as one group of 10

(5 red and 5 black), and three more.

5 2



Constructing a Rekenrek

• What do we need?• A small cardboard rectangle

• String

• 20 beads



• Tie a knot at the end of each string.

• Slip the ends of the string through the slits on the

cardboard so that the beads are on the front of

the cardboard, and the knot of the string is on theback side.

Constructing a Rekenrek



Flash Attack• Objectives

– To help students begin to “subitize” -- to see acollections of objects as one quantity rather thanindividual beads

– To help students develop visual anchors around 5

and 10 – To help students make associations between

various quantities. For example, consider the way apupil might make the connection between 7 and 10.

“I know there are ten beads in each row. Therewere three beads left in the start position. So…there must be 7 in the row because 10 - 3 = 7.”



Examples• Time two secs (so they do not have time to count each bead

individually)

• Push over beads and ask “How many? and ask for

explanations.

Listen for …”I knew there were reds, and /less onemore” (encourage this identification of 5 reds/ ten beads as

central to the solution)

• Expand to two rows. Ask “How many beads are there in the

top row? The bottom row? Altogether? (Reasoning)

• Look for patterns as they determine the quantities being

flashed (how they know without counting each bead)



Basic Combinations 0-10• Modeling the activity:

– Use both rows to keep the addends clearly visible – Suggested sequence (begin with 5 on the top row)

• “Let’s make 8. I start with 5. How many more?”



• “Let’s make 6 again. I start with 3. How many more?”

– E.g., Build on previous relationship. Use doubles.

– Listen to the thinking/explanations of students.

– Expand to Combinations of 10-20

Activity:

Extend to Subtraction



Basic Combinations 0-10

• Lesson Objectives

– Relational View of Equality

• One of the strengths of the rekenrek is its connections to

other forms of mathematical reasoning. For example,

equality .• Our children are only used to seeing problems like…

» 4 + 5 = ? 6 + 3 = ?

– Part-Part-Whole relationships

– Missing Addend problems

– Continue to build informal strategies and means for

combining numbers



Doubles• Objectives

– Help students visualize doubles (e.g., 4+4; 6+6)

– Help students use doubles in computation

• The visualization is key:

1+1=2 2+2=4 3+3=6 4+4=8



Developing Understanding of the Doubles

– Ask students what they notice about these visualizations.

They might see them as vertical groups of two… as two

horizontal lines of the same number of beads… evennumbers… etc.

– As students are ready, teachers should include the doubles

between 6 and 10, following a similar teaching strategy. In

this case, students should know to use their knowledge of adouble of 5 (two groups of 5 red beads = 10) to compute

related doubles.



7 + 7 = 14

Seen as, perhaps…

2 groups of 5, plus 4



Almost a Double • Students should use their understanding of

doubles to successfully work with “near doubles”,

e.g., 7 + 8

• Students can begin to recognize the difference

between even numbers (even numbers can be

represented as a pair of equal numbers) and oddnumbers (paired numbers plus one).



Near Doubles…

1 2+1 = 3 4+1 = 5 6+1 = 7 8+1 = 9

2 + 1 3 + 2 4 + 3 5+ 4

Use a pencil

to separate



Near Doubles: Developing Ideas

• Develop this idea by doing several additionalexamples with the Rekenrek. Ask pupils to use theRekenrek to “prove” whether or not the following aretrue.

• Have students visually identify each component ofthese statements:

• Does 6 + 7 = 12 + 1?

• Does 3 + 2 = 4 + 1? • Does 4 + 5 = 8 + 1? • Does 8 + 9 = 16 + 1?



Part-Part-Whole

• To develop an understanding of part-part-wholerelationships in number problems involvingaddition and subtraction.

• To develop a relational understanding of theequal sign.

• To develop confidence and comfort with“missing addend” problems.



Developing Ideas• Push some beads to the left. Cover the remaining beads.

• Ask students: “How many beads do you see on the top row?”

• Ask: “How many beads are covered (top row)?”

• Listen for answers like the following: – “5 and 1 more is 6. I counted up to 10 … 7, 8, 9, 10.

4 are covered.”

– “I know that 6 + 4 = 10. I see 6, so 4 more.”

– “I know that there are 5 red and 5 white on each row. I only see onewhite, so there must be 4 more.”

• Next… move to both rows of beads.

Cover

Remaining

Beads



Show the Sum: How many ways?

• Get two numbers (maybe from the rolling of two dice)..

• Teacher: “The dice turned up 9 and 6. What is the sum?”

• Get students to share their strategies for calculating the sum.

• Eg.: I moved 5 red and 4 white on the top to show 9. Since Iknow 5 plus 1 is 6, on the bottom I moved all 5 red beads andone more white one. They all add up to 15.

• (Record the number sentences to represent each student’sstrategy)

A t ti it



Automaticity:

Math Facts Objectives in Learning the Math Facts

Quick recall, yes. But… with understanding, and with astrategy!

• To develop fluency with the addition number facts through 20.

• To reinforce anchoring on 10 and using doubles as helpful strategiesto complete the math facts through 20.

• Students can model the number facts on one row of the rekenrek

(like 5 + 4), or model facts using both rows (which they have to do

when the number get larger, e.g., 8 + 7)



• To promote automaticity, it is important to use

the Rekenrek to visually represent the numberfacts for children.

• It is important to reinforce the various strategies

that the pupils used.



• With other number facts, other compensation strategies

should be encouraged. E.g: 8 + 4 = 12

• While some students might see doubles, it is more likelyin this instance that pupils will anchor on 10 by mentally

sliding two additional beads to the leaft on the top row,

and then compensating on the bottom row by removing

two beads.



Tens or Ones

• Draw one of the number cards.

• Ask them to build the number on their rekenrek

• Next draw from the second container (the Take

Away Tens and Run Away Ones cards)

• Perform the action and look for different

strategies and invite student sharing.

• Let’s try





The Rekenrek and a Relational

View of Equality

• Take, for example, the = sign

What belongs in the box?

8 + 4 = + 5

How do children often answer this problem?

Discuss in pairs… (Activity)



8 + 4 = + 5

• 3 research studies used this exact problem• No more than 10% of US students in grades 1-6 in

these 3 studies put the correct number (7) in the box.In one of the studies, not one 6th grader out of 145 put

a 7 in the box.• The most common responses?

• 12 and 17

• Why?• Students are led to believe through basic factexercises that the “problem” is on the left side, and the“answer” comes after the = sign.

• Rekenrek use in K 3 mitigates this misconception

Rekenrek (Eng Version

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